Fundamental Theorem of Calculus

Fundamental Theorem of Calculus: If $f$ is Riemann integrable on $[a,b]$ and $F$ is a function on $[a,b]$ for which $F^{\prime }(x)=f(x)$ for all $x\in [a,b],$ then

$${\int }_a^{b}f(x)dx=F(b)-F(a).$$

Derivative Part of the Fundamental Theorem: Let $f$ be continuous on $[a,b]$ and define the function $F$ on $[a,b]$ by

$$F(x)={\int }_a^{b}f(y)dy.$$

Then, $F$ is continuous on $[a,b],$ and for all $x\in (a,b),$ $F$ is differentiable and $F^{\prime }(x)=f(x).$

Integration by Parts Theorem: Let $f$ and $g$ be continuously differentiable functions on $[a,b].$ Then

$${\int }_a^{b}f(x)g^{\prime }(x)dx=f(b)g(b)-f(a)g(a)-{\int }_a^b{f}^{\prime }(x)g(x)dx.$$

Substitution Theorem for Integration: Let $g$ be a function with a continuous derivative on $[a,b].$ If $f$ is continuous on the range of $g,$ then

$${\int }_a^{b}f(g(x))g^{\prime }(x)dx={\int }_{g(a)}^{g(b)}f(u)du.$$